A discrete-time GeoX/G/1 retrial queueing system with starting failures and optional service

This paper is concerned with the analysis of a single-server batch arrival retrial queue with starting failures in discrete time in which all the arriving customers need essential service (ES) necessarily whereas only some may require the optional service (OS) after ES. The customers are assumed to arrive at the system according to a geometric process during the consecutive slots. The service times and retrial times are generally distributed. The generating functions of the number of customers in the retrial group and in the system are obtained along with the marginal distributions of the number of customers in the retrial group when the server is idle, busy (in providing ES or OS) or broken down. The explicit formulae for the stationary distribution and other performance measures at equilibrium are obtained. Some special cases are deduced by setting appropriate parameters. The stochastic decomposition property has also been discussed. The numerical examples are facilitated to explore the effect of different parameters on various performance measures.